#496952

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22135
prime radical	#3067	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,12,7,13,5,30,9,15,32,42,6,14,17,41)(3,16,21,46,22,36,19,52,23,43,54,62,20,10,33,35)(4,24,27,57,28,40,31,38,29,44,58,56,26,11,25,39)(18,34,50,60,49,45,53,64,51,37,61,59,48,55,47,63);(1,3,17,33,6,20,32,54,9,23,5,19,7,22,8,21)(2,10,30,46,14,43,12,35,15,36,41,62,13,16,42,52)(4,18,25,47,26,48,58,61,29,51,31,53,28,49,27,50)(11,37,57,63,44,45,39,59,40,34,56,64,24,55,38,60);(1,4)(2,11)(3,18)(5,31)(6,26)(7,28)(8,27)(9,29)(10,34)(12,39)(13,24)(14,44)(15,40)(16,45)(17,25)(19,53)(20,48)(21,50)(22,49)(23,51)(30,57)(32,58)(33,47)(35,60)(36,37)(38,42)(41,56)(43,55)(46,64)(52,59)(54,61)(62,63);(1,5,6,8,9,17,7,32)(2,12,13,30,15,42,14,41)(3,19,20,21,23,33,22,54)(4,25,26,58,29,31,28,27)(10,35,16,46,36,52,43,62)(11,38,24,56,40,39,44,57)(18,47,48,61,51,53,49,50)(34,59,45,63,37,60,55,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=ceh,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224954, #72224960, #72224961, …
maximal normal subgroup of	#72224954, #72224960, #72224961, …
maximal quotient of	C₂×#496952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404148);